I would use double buffering and a Read/Write lock.
Double buffering reduces holdups by allowing processing of the swapped-out map.
Using a Read/Write lock allows me to be certain that no-one is still writing to the map after we swap.
class DoubleBufferedMap<K, V> extends AbstractMap<K, V> implements Map<K, V> {
    // Used whenever I want to create a new map.
    private final Supplier<Map<K, V>> mapSupplier;
    // The underlying map.
    private volatile Map<K, V> map;
    // My lock - for ensuring no-one is writing.
    ReadWriteLock readWriteLock = new ReentrantReadWriteLock();
    final Lock readLock = readWriteLock.readLock();
    final Lock writeLock = readWriteLock.writeLock();
    public DoubleBufferedMap(Supplier<Map<K, V>> mapSupplier) {
        this.mapSupplier = mapSupplier;
        this.map = mapSupplier.get();
    }
    /**
     * Swaps out the current map with a new one.
     * 
     * @return the old map ready for processing, guaranteed to have no pending writes.
     */
    public Map<K,V> swap() {
        // Grab the existing map.
        Map<K,V> oldMap = map;
        // Replace it.
        map = mapSupplier.get();
        // Take a write lock to wait for all `put`s to complete.
        try {
            writeLock.lock();
        } finally {
            writeLock.unlock();
        }
        return oldMap;
    }
    // Put methods must take a read lock (yeah I know it's weird)
    @Nullable
    @Override
    public V put(K key, V value) {
        try{
            // Take a read-lock so they know when I'm finished.
            readLock.lock();
            return map.put(key, value);
        } finally {
            readLock.unlock();
        }
    }
    @Override
    public void putAll(@NotNull Map<? extends K, ? extends V> m) {
        try{
            // Take a read-lock so they know when I'm finished.
            readLock.lock();
            map.putAll(m);
        } finally {
            readLock.unlock();
        }
    }
    @Nullable
    @Override
    public V putIfAbsent(K key, V value) {
        try{
            // Take a read-lock so they know when I'm finished.
            readLock.lock();
            return map.putIfAbsent(key, value);
        } finally {
            readLock.unlock();
        }
    }
    // All other methods are simply delegated - but you may wish to disallow some of them (like `entrySet` which would expose the map to modification without locks).
    @Override
    public Set<Entry<K, V>> entrySet() {
        return map.entrySet();
    }
    @Override
    public boolean equals(Object o) {
        return map.equals(o);
    }
    @Override
    public int hashCode() {
        return map.hashCode();
    }
    // ... The rest of the delegators (left to the student)